Question

the life-span of every star is related to mass, which determines luminosity (brightness). the table shows the masses, lifetimes, and luminosities of two stars compared to that of the sun. how does the sun’s lifetime compare to other main sequence stars? move each lifetime into the table next to the star it describes.

star	mass (m⊙)	main sequence lifetime (years)	luminosity (l⊙)
spica	11		2200
sun	1		1

10⁷ 10¹⁰ 10¹²

Explanation:

Step1: Recall the relationship between mass, luminosity, and lifetime of stars.

The main sequence lifetime of a star is approximately proportional to $\frac{\text{Mass}}{\text{Luminosity}}$ (since lifetime $\approx \frac{\text{Available Fuel (proportional to mass)}}{\text{Rate of Fuel Consumption (proportional to luminosity)}}$). For the Sun, let's denote its mass as $M_{\odot}=1$, luminosity as $L_{\odot}=1$, and lifetime as $t_{\odot}\approx 10^{10}$ years (a known value for the Sun's main sequence lifetime).

Step2: Calculate the lifetime of Proxima Centauri.

For Proxima Centauri, $M = 0.12M_{\odot}$ and $L = 0.0017L_{\odot}$. Using the lifetime formula $t\approx\frac{M}{L}t_{\odot}$, we substitute the values:
$t_{\text{Proxima}}=\frac{0.12}{0.0017}\times 10^{10}\approx 7.06\times 10^{11}\approx 10^{12}$ (approximate, since $\frac{0.12}{0.0017}\approx 70.59$, and $70.59\times 10^{10}=7.059\times 10^{11}\approx 10^{12}$ when considering order of magnitude).

Step3: Calculate the lifetime of Spica.

For Spica, $M = 11M_{\odot}$ and $L = 2200L_{\odot}$. Using the same formula:
$t_{\text{Spica}}=\frac{11}{2200}\times 10^{10}=\frac{1}{200}\times 10^{10}=5\times 10^{7}\approx 10^{7}$ (since $\frac{11}{2200}=0.005$, and $0.005\times 10^{10}=5\times 10^{7}$).

Step4: Assign lifetimes to stars.

• Proxima Centauri: Since its lifetime is much longer than the Sun's (calculated to be around $10^{12}$ years), we assign $10^{12}$.
• Spica: Its lifetime is much shorter than the Sun's (calculated to be around $10^{7}$ years), we assign $10^{7}$.
• Sun: The Sun's main sequence lifetime is approximately $10^{10}$ years, so we assign $10^{10}$.

Answer:

Star	Mass ($M_{\odot}$)	Main Sequence Lifetime (years)	Luminosity ($L_{\odot}$)
Spica	11	$10^{7}$	2200
Sun	1	$10^{10}$	1